From/to airport
Capodichino airport (NAP) is very close to the city, and served by a special bus ride (quite short in duration). Name of the bus is " Alibus ", ride is 5 Euros (you can buy tickets at special machines in the airport hall or on board, paying the driver) and you will find it some 130 meters/yards from the arrival exit. Cross the small square full of taxis and proceed along the lane, you will see a small square with a very official looking bus. Please beware of illegals and taxy drivers that will try to lure you with cries "Alibus. Alibus." The official one has an official look (and it is plastered in advertisements) serves only one stop in the square in the airport, and two stops (central station + dock x 2) in town.
Taxis
We strongly discourage their use, unless you really need them (we realize in some case they may be necessary). The official ones should be either white or yellow (very few of them) and have a service identity code (usually name of a city followed by some numbers) on their side doors. It is a good idea to take a picture of their ID number (and plate, if possible) before getting on board. This should discourage the driver from keeping the taxi-meter shut or other (alas, common) malpractices. At any rate, check that the taxi-meter starts running when the car starts, and complain if they keep it shut. A ride from the airport to anywhere in town should not cost more than 25 Euros.
Metro
The metro Line 1 connects the Central Station (Piazza Garibaldi) to the rest of the city, running through the historical center (downtown, see below). It is safe and cheap, yet its frequency is not the highest, and it close rather early at night, around 11pm. If you are to move only within the downtown, we encourage you to use good old footwork: up to three-four metro stations may cost you in wait longer than it takes to walk from one point to the other.
Uphill, "Vomero" neighborhood
If you take the good decision to explore the gems on the hill Vomero, we encourage you not to use the metro, but rather one of the two currently open "Funicolare". Funicolare Centrale, or Funicolare di Montesanto, at least for the way up. In both cases you have to get out at the upmost, last stop (the train won't be able to move farther, and will go back after 5 minutes). They are cheap (one way ticket 1.3 Euros), efficient, fast. Only, they close at 10pm.